A Level Further Mathematics 9231 Paper 32, Oct/Nov 2024

A particle of mass $2\text{ kg}$ is attached to one end of a light elastic string of natural length $0.8\text{ m}$ and modulus of elasticity $100\text{ N}$. The other end of the string is attached to a fixed point $O$ on a smooth horizontal surface. The particle is moving in a horizontal circle about $O$ with the string taut and with constant angular speed $5\text{ radians per second}$. Find the extension of the string.

A particle $P$ of mass $m$ is attached to one end of a light elastic spring of natural length $a$ and modulus of elasticity $5mg$. The other end of the spring is attached to a fixed point $O$. The spring hangs vertically with $P$ below $O$. The particle $P$ is pulled down vertically and released from rest when the length of the spring is $\frac{3}{2}a$. Find the distance of $P$ below $O$ when $P$ first comes to instantaneous rest.

The diagram shows two identical smooth uniform spheres $A$ and $B$ of equal radii and each of mass $m$. The two spheres are moving on a smooth horizontal surface when they collide with speeds $2u$ and $3u$ respectively. Immediately before the collision, $A$’s direction of motion makes an angle $\theta$ with the line of centres and $B$’s direction of motion is perpendicular to that of $A$. After the collision, $B$ moves perpendicular to the line of centres. The coefficient of restitution between the spheres is $\frac{1}{3}$.

The end $A$ of a uniform rod $AB$ of length $6a$ and weight $W$ is in contact with a rough vertical wall. One end of a light inextensible string of length $3a$ is attached to the midpoint $C$ of the rod. The other end of the string is attached to a point $D$ on the wall, vertically above $A$. The rod is in equilibrium when the angle between the rod and the wall is $\theta$, where $\tan \theta =\frac{3}{2}$. A particle of weight $W$ is attached to the point $E$ on the rod, where the distance $AE$ is equal to $ka$ ($3<k<6$) (see diagram). The rod and the string are in a vertical plane perpendicular to the wall. The coefficient of friction between the rod and the wall is $\frac{1}{3}$. The rod is about to slip down the wall.

A particle $P$ is projected from a point $O$ on horizontal ground with speed $u$ at an angle $\theta$ above the horizontal, where $\tan \theta =\frac{1}{3}$. The particle $P$ moves freely under gravity and passes through the point with coordinates $(3a,\frac{4}{5}a)$ relative to horizontal and vertical axes through $O$ in the plane of the motion.

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held with the string taut and the string makes an angle $\theta$ with the downward vertical through $O$. The particle $P$ is projected at right angles to the string with speed $\frac{1}{3}\sqrt{10ag}$ and begins to move downwards along a circular path. When the string is vertical, it strikes a small smooth peg at the point $A$ which is vertically below $O$. The circular path and the point $A$ are in the same vertical plane. After the string strikes the peg, the particle $P$ begins to move in a vertical circle with centre $A$. When the string makes an angle $\theta$ with the upward vertical through $A$ the string becomes slack (see diagram). The distance of $A$ below $O$ is $\frac{5}{9}a$.

A particle $P$ of mass $m\text{ kg}$ is held at rest at a point $O$ and released so that it moves vertically under gravity against a resistive force of magnitude $0.1m{v}^2\text{ N}$, where $v{\text{ ms}}^{-1}$ is the velocity of $P$ at time $t\text{ s}$.